Need up to 30 seconds to load.

hey folks boy am I tired thankfully I've

got a viewer-requested lesson for you

today so I don't have to think of

anything

Cooper 2:5 said I'm working on a math

problem and it only gave me the area of

the square no width or length and it

asks me what's the length of the

diagonal then he has a bunch of ominous

skulls and says oh okay well because you

put it in such an intimidating manner

I'm here will solve the problem let's go

through it together

remember the problem is we are given the

area of a square so let's go ahead and

start with a drawing of our square like

any good geometry problem we got our

square and we're given the area of the

square for this example let's say the

area of the squares twenty-five

twenty-five the unit's don't matter

it could be square centimeters square

inches I'm just going to write you

squared so 25 units squared whatever the

unit is doesn't matter we want to find

the length of the diagonal from the

square given the area the diagonal of

course looks something something like

that now immediately upon drawing the

diagonal just to start visualizing our

problem you should get some hint of how

we can solve this problem remember that

all of the angles in a square are right

angles so a particular interest this one

here is a right angle so by drawing this

diagonal we've just defined a right

triangle so then you might think does

Pythagorean theorem is that going to

help us out here and as it turns out it

will as it often does now the way your

your thought process might go to solve

this problem is oK we've got the area of

a square we want to find the length of

the diagonal so do we know have we

solved the similar problem with the same

unknown the unknown being the length of

the diagonal of a square and one that

might come to mind is a problem where

we're given a side length of the square

and we want to find the diagonal from

that information of course all of the

sides of a square have the same length

so if this side is s this side is also s

and look at that to find the diagonal

from this information we just use the

theorem s squared plus s squared equals

d squared but a bing bada boom there's

your answer so if we can get the side

length of the square from the area then

we're back in familiar territory a

problem that we've all probably already

done so can we get the side length from

the area of course we can because the

area of a square is just the square of

its side length so if the side length is

s the area is s squared so in particular

we have that the wall right this s

squared is equal to 25 units squared and

just take the square root of both sides

and that will give us our side length

and then we can go ahead and use the

Pythagorean theorem so we have that the

side length s is equal to the square

root of 25 units squared which is just 5

units so then we can erase these s's we

know what S is equal to now the side

lengths of our square five remember all

the side lengths are the same so they're

all five we could go ahead and throw you

in there so it's five units five units

and we just use the Pythagorean theorem

remember the Pythagorean theorem tells

us that the sum of the squares of the

legs of a right triangle is equal to the

hypotenuse squared so we've got that

five units squared plus five units

squared this is all getting squared five

units squared five units squared is

equal to the square of the hypotenuse

which is the diagonal in this case again

that's the sum of the squares of the

legs is equal to the square of the

hypotenuse five units squared is 25

units squared we already saw that up

here so 25 units squared plus 25 units

squared that's 50 units squared and so

we're almost done we've got that 50

units squared is equal to d squared just

take the square root of both sides here

and we've got that the diagonal D is

equal to the square root of 50 units

squared so we could just write that as

the square root of 50 and then the

square root of units square

is units so this is just the square root

of 50 units and that units is outside of

the square root and that is an exact

answer so if the area was 25 inches

squared

then the diagonal is root 50 inches and

we could approximate this if we wanted

to it's about seven point zero seven

units just pausing here for a moment to

point out that this same exact procedure

would give us the same exact result if

we were trying to find the other

diagonal of the square the one that goes

the other way all right on to the rest

of the lesson so I want you to try this

one on your own before watching the

solution let's say we're given that the

area of the square is 16 and we'll say

we'll work with an actual unit this time

so say 16 centimeters squared so go

ahead solve this problem just like we

did before to find the diagonal of a

square that has an area of 16

centimeters squared all right hopefully

you've got an answer now when I go

through this problem I'm going to leave

out the units once you do a couple of

these problems you know that if you're

given an area in centimeters squared for

example your side length is going to be

in centimeters if you're given your area

in inches squared your side length is

going to be in inches and so on so we

can leave off the units we know what the

units will be at the end so how do we

solve this problem same way we did

before the area is 16 centimeters

squared which is the square of the side

length of the square so then we just

take the square root of both sides

square root of that square root of that

so leaving off our units we have that

the side length of the square is equal

to the square root of 16 what's the

square root of 16 well of course that's

4 so then we can go ahead and erase our

s's over here we know that the side

length of this square are all four now

we've got a right triangle with leg

lengths of four we just go ahead and

apply our favorite Pythagorean theorem

that tells us that the sum of the

squares of the leg lengths four squared

plus four squared that's the sum of the

squares of the

leg lengths is equal to the diagonal

squared d squared the hypotenuse which

in this case is our diagonal that we're

looking for 4 squared plus 4 squared

that's 16 plus 16 and that is 32 so 4

squared plus 4 squared 16 plus 16 that's

32 is equal to the diagonal squared take

the square root of both sides and we've

got our answer if you liked the video of

the square is equal to the square root

of 32 so since our area was given in

centimeters squared

we know that this diagonal is the square

root of 32 centimeters long and if you

approximated this square root you would

have got about five point six six

centimeters and that's all there is to

it given the area of a square the square

root of that is the side length of the

square those are our legs of this right

triangle and then we just apply the

Pythagorean theorem side length squared

plus side length squared is equal to the

hypotenuse squared and in this case the

hypotenuse is our diagonal and of course

the answer falls out immediately from

that if you haven't seen a proof of the

Pythagorean theorem by any chance I'll

leave a link in the description to a

lesson where I prove the Pythagorean

theorem and even if you're not big on

the geometry and the proofs I'm eating

bugs the whole time so you might find it

a little bit more amusing in any event I

hope this video helped you understand

how to find the length of a diagonal

from the square when given the area be

sure to let me know in the comments if

you have any questions need anything

clarified or have any other video

requests thank you very much for

watching I will see you next time and be

sure to subscribe for the swankiest math

lessons on the Internet

[Music]

[Music]